On the density of critical graphs with no large cliques
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A graph G is k-critical if χ(G) = k and every proper subgraph of G is (k - 1)-colorable, and if L is a list-assignment for G, then G is L-critical if G is not L-colorable but every proper induced subgraph of G is. In 2014, Kostochka and Yancey proved a lower bound on the average degree of an n-vertex k-critical graph tending to k - 2/k - 1 for large n that is tight for infinitely many values of n, and they asked how their bound may be improved for graphs not containing a large clique. Answering this question, we prove that for ε≤ 2.6·10^-10, if k is sufficiently large and G is a K_ω + 1-free L-critical graph where ω≤ k - log^10k and L is a list-assignment for G such that |L(v)| = k - 1 for all v∈ V(G), then the average degree of G is at least (1 + ε)(k - 1) - εω - 1. This result implies that for some ε > 0, for every graph G satisfying ω(G) ≤mad(G) - log^10mad(G) where ω(G) is the size of the largest clique in G and mad(G) is the maximum average degree of G, the list-chromatic number of G is at most (1 - ε)(mad(G) + 1) + εω(G).
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